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A solution to the energy minimization problem constrained by a density function


Author: Kanya Ishizaka
Journal: Math. Comp. 86 (2017), 275-314
MSC (2010): Primary 46N10, 52C35; Secondary 49N45, 26A51
DOI: https://doi.org/10.1090/mcom/3136
Published electronically: April 26, 2016
MathSciNet review: 3557800
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Abstract: We present a new solution to the problem of determining an energy integral which has a unique minimum at a given Borel probability measure on a compact metric space. For a continuous kernel, we show that there exists a unique weight function such that the given measure is an equilibrium measure with respect to the kernel multiplied by the weight function. The weight function is determined as a unique fixed point of a functional operator. Moreover, if the kernel satisfies the energy principle on the space, then the given measure achieves a unique minimum of the energy integral with respect to the weighted kernel. In order to obtain a kernel satisfying the energy principle on Euclidean subspaces, we improve the condition shown by Gneiting for a defining function of a kernel to belong to the Mittal-Berman-Gneiting class. By using the obtained condition, we show related results for the energy with the kernel. Finally, we present practical examples of distributing a finite number of points that are constrained by a density function.


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Kanya Ishizaka
Affiliation: Key Technology Laboratory, Research & Technology Group, Fuji Xerox Co., Ltd., 430 Sakai, Nakai-machi, Ashigarakami-gun, Kanagawa, 259-0157, Japan
Email: Kanya.Ishizaka@fujixerox.co.jp

DOI: https://doi.org/10.1090/mcom/3136
Keywords: Minimum energy state, optimal distribution, density constraint, positive definite kernel, energy principle, halftoning
Received by editor(s): November 22, 2012
Received by editor(s) in revised form: July 10, 2015
Published electronically: April 26, 2016
Article copyright: © Copyright 2016 American Mathematical Society